There are several significant ways in which spinors and twistors impinge on conformal differential geometry. The isomorphisms give a powerful computational tool for differential geometry in six dimensions, just as do spinors in four dimensions. Part of the special utility arises from the fact that in six dimensions spinors are automatically pure, whereas this is not the case in any higher dimension. The isomorphism also underpins the use of twistors in four-dimensional conformal geometry, the point being that SO is the group of conformal motions of compactified Minkowski space. The local twistor connection may be defined quite explicitly in terms of the Levi-Civita connection of a metric in the conformal class. The interest in foliations of space-times by such surfaces goes back in effect more than three decades to 1961, when Ivor Robinson published a result that was to have a significant influence on the subsequent development of general relativity.